DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION. Keenan Crane CMU (J) Spring 2016

Transcription

1 DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU (J) Spring 2016

2 PART III: GEOMETRIC STRUCTURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU (J) Spring 2016

3 Hierarchy of Structure more rigid floppier topology differential topology geometry

4 Geometric Structure (differentiable structure) volumes angles (volume form) (complex structure) geometry (metric)

5 Orientation Visualized

6 Determinant Review Q: What does it mean?

7 Determinant Visualized Key idea: determinant is change in (signed) volume.

8 Determinant

9 Orientation-Preserving Map (Linear) orientation-preserving orientation-reversing

10 Orientation-Preserving Map (Differentiable)

11 Oriented Manifold orientationpreserving Key idea: M now has one of two orientations.

12 Non-Orientable Manifold Not every differentiable manifold is orientable!

13 Oriented Simplicial Complex Example.

14 Volume Form Antisymmetric: swapping any pair of arguments flips the sign of the result. Multilinear: linear in each argument (keeping all others fixed). Nonvanishing: nonzero for linearly independent arguments. v Key idea: measures (oriented) volume spanned by vectors. p u

15 Motivation: Integration Z v i p i i  w pi (u i, v i ) =) W w u i W

16 Volume Form Example (See Reading 4.)

17 Pushforward and Pullback of Functions

18 Pushforward and Pullback of Volume Form f df (v) v df (u) u

19 Volume-Preserving Diffeomorphism Important example: Motion of incompressible fluids.

20 Volume Conservation in Fluid Simulation poor volume conservation good volume conservation from Lentine et al, Simulating Free Surface Flow with Very Large Time Steps

21 Moser s Theorem Application: Optimal Transport pushforward Solomon et al, Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains

22 Discrete Volume Form? How do we encode volume on a simplicial manifold? First attempt: volume per simplex. (E.g., area per triangle.) Q: Are there motions of (interior) vertices that preserve volume of all simplices? preserve triangle areas? A: No, not in general. E.g., simplicial disk: (Euler s polyhedral formula) (B = #boundary edges) Constraints: Degrees of freedom: More constraints than DOFs. (Argument courtesy U. Pinkall.)

23 Poincaré Dual (2D) primal dual Q: Where is the center of a triangle?

24 Oreochromis Mossambicus (a.k.a. Voronoi Fish )

25 Pomegranate (a.k.a. Voronoi Fruit )

26 Voronoi Diagrams Voronoi lizard Voronoi bird Voronoi McDonald s Voronoi foam Voronoi cookies Voronoi printing

27 Voronoi Diagram / Delaunay Triangulation Voronoi diagram (Poincaré Duality) Delaunay Triangulation

28 Triangle Centers circumcenter barycenter incenter Voronoi/Delaunay Fact: There are infinitely many triangle centers.

29 Discrete Volume-Preserving Diffeomorphisms Several possibilities: barycentric dual [Irving et al 2007] circumcentric, incentric dual [???] power diagram [de Goes et al 2015] doubly-stochastic matrices [Pavlov et al 2009]

30 Conformal Geometry Visualized

31 Complex Numbers nonsense! More importantly: obscures geometric meaning.

32 Imaginary Unit Geometric Description Symbol ι denotes quarter-turn in the counter-clockwise direction.

33 Complex Arithmetic Visualized * rectangular coordinates addition multiplication i <-> 1

34 Complex Product Euler s identity: * (Now forget the algebra and remember the geometry!)

35 Holomorphic Map Visualized Key idea: angles & orientation are preserved; scale may change (uniformly!).

36 Holomorphic Map Q: Why are these descriptions equivalent? Q: Why are arbitrary angles preserved? Q: Is orientation preserved?

37 Characterizations of Holomorphic Maps c In general: holomorphic functions more rigid than differentiable ones (more later!)

38 Conformal Maps on Surfaces Visualized not conformal conformal

39 Complex Structure holomorphic Key idea: we now have a notion of angles on M.

40 Almost Complex Structure

41 Riemann Surface Q: Can you think of a nonorientable surface (like the Klein bottle) as a Riemann surface?

42 Holomorphic Maps from a Surface to the Plane

43 Example Stereographic Projection

44 Reminder: Linear Transformations via Invariants Transformation Invariant LINEAR MAP lines through origin LINEAR ISOMORPHISM dimension ROTATION length & orientation REFLECTION length & reverses orientation UNIFORM SCALE everything but scale!

45 Example Möbius Transformations (2D)

46 Möbius Transformations Revealed

47 Möbius Transformations (nd) Sphere inversion: (plus rotation, translation & uniform scale) Q: What happens to orientation?

48 Liouville s Theorem Key idea: conformal structure is quite rigid in dimension 3.

49 Riemann Mapping Theorem conformal Möbius Riemann map Riemann map

50 Flexibility of Conformal Maps in 2D conformal Riemann map Riemann map SURFACE TO PLANE PLANE TO PLANE Key idea: conformal maps quite flexible in dimension 2. SURFACE TO SURFACE

51 Discrete Conformal Map? How should we discretize conformal (i.e., angle-preserving) maps in 2D? First attempt: preserve angles of each triangle: similarity

52 Rigidity of Angle Preservation Mesh is rigid (up to a single global scale & Euclidean motion)!

53 Discrete Conformal Maps Many different ideas for discrete conformal invariants: 1. angles per triangle (e.g., angle-based flattening [Sheffer & de Sturler 2001]) 2. zero shear per triangle (e.g., least-squares conformal maps [Lévy et al 2002]) 3. dual/primal edge length ratio (e.g., discrete Riemann surfaces [Mercat 2001]) 4. circle intersection angles (e.g., circle patterns [Kharevych et al 2006]) 5. length cross ratio (e.g., conformal equivalence of triangle meshes [Springborn et al 2008]) 6. inversive distance (e.g., weighted triangulations [Guo et al 2009]) 7. facewise Möbius transformations (e.g., conformal mesh deformations [Vaxman et al 2015]) RIGID (Note: rigid schemes are nonetheless still useful!) FLEXIBLE *(7) generalizes both (4) and (5). Thanks to F. de Goes for useful discussions.

54 Conformal Equivalence of Triangle Meshes Springborn, Schröder, Pinkall, Conformal Equivalence of Triangle Meshes length cross ratio discrete conformal equivalence Key idea: just as flexible as smooth maps!

55 Möbius Invariance of CETM Fact. Length cross ratios are exactly preserved by Möbius transformations of vertices (even though angles are not!) Möbius Möbius Möbius Key idea: discrete theory may not always capture most obvious properties; must often think more broadly: what are other characterizations/structures/invariants?

56 Riemannian Structure Visualized

57 Volume vs. Angle Preservation VOLUME-PRESERVING ANGLE-PRESERVING LINEAR NONLINEAR MANIFOLD

58 Area + Angle Preserving Maps Q: Which linear maps are area and angle preserving? A: Rotations & reflections. Q: Which ones preserve area, angle, and orientation? A: Just rotations.

59 Length, Angle, and Area Q: Why does preservation of lengths imply preservation of angles? A: Side-side-side! a b c b c a Q: Why does preservation of lengths imply preservation of areas? A: Heron s formula. a b c (Can you argue geometrically, without appeal to authority?)

60 Inner Product Q: What s the geometric motivation for this definition?

61 Inner Product Geometric Motivation Q: Geometric interpretations of inner product? PROJECTION ANGLE Q: Where can you see symmetry? Linearity? Positivity? Definiteness?

62 Euclidean Isometry isometry a b c b c Key idea: Euclidean motions are extremely rigid! a

63 Reflection Puzzle Q: Why do objects in a mirror get reversed left/right rather than top/bottom?

64 Flat Manifold Euclidean isometry Key idea: lengths, angles, areas all preserved.

65 Classification of (Compact) Flat 2-Manifolds PLANE CYLINDER MÖBIUS BAND TORUS KLEIN BOTTLE PROJECTIVE PLANE + 15 quotients by discrete groups (orbifolds)

66 Example Flat Torus Key idea: degree of differentiability matters! Borelli, Jabrane, Lazarus, Thibert, Isometric Embeddings of the Square Flat Torus in Ambient Space

67 Riemannian Manifold Key idea: can now measure angle and length of tangent vectors.

68 Metric From Complex Structure & Volume Form SYMMETRY POSITIVE-DEFINITENESS LINEARITY Yes, by definition. (geometry?) Key idea: metric encodes area and angle (and vice versa).

69 Complex Structure & Volume Form from Metric Q: Can we go the other way, and recover volume/angle from the metric? A: Almost, but we also need to pick an orientation! VOLUME (two choices of root) ANGLE (two choices of sign)

70 Conformal Structure Metric Characterization

71 Metric from Immersion (No geometry! Just topology & smooth structure.)

72 Discrete Metric

73 Discrete Metric from Simplicial Immersion Key idea: discrete metric is simply the induced edge lengths!

74 Isometry isometry isometry near-isometry Notice: local Euclidean isometry doesn t necessarily mean globally flat!

75 Isometric Rigidity (smooth & discrete!) Not necessarily true for nonconvex polyhedra. E.g., Connelly s flexible sphere: More generally? Full story about metric flexibility remains unwritten. General wisdom: for closed surfaces, perfect isometry is pretty darn rigid.

76 Length of a Curve

77 Geodesic (symmetry? positive-definiteness? triangle inequality?)

78 Myers Steenrod Q: Every Riemannian metric induces a point-to-point distance metric, induced by geodesic distance. But does every point-to point distance also determine a Riemannian metric? A: Basically, yes. (See Palais, On the Differentiability of Isometries ).

79 Gaussian Curvature Visualized

80 Scalar Curvature

81 Discrete Gaussian Curvature Fact. Edge lengths determine corner angles (law of cosines)

82 Discrete Gaussian Curvature Motivation From our general definition of scalar curvature, we have Substitute area of Euclidean ball area of geodesic wedge area of geodesic ball Then

83 Cone Metric Visualized

84 Discrete Gauss-Bonnet g = 0 g = 1 g = 2

85 Proof of Discrete Gauss-Bonnet (Intrinsic)

86 Discrete Metric via Weighted Triangulation Idea: represent surface as triangles and weights per vertex Adds additional flexibility / accuracy / preservation of structure (Note: different object of study no longer studying geometry of pure triangle meshes!) de Goes et al, Weighted Triangulations for Geometry Processing

87 Structure from Data Ongoing question in (D)DG: what data encodes different structures? rigidity / flexibility simplicity computability # "Columbia Banana" # Keenan Crane - January 25, 2014 # # VERTICES: 107 # EDGES: 315 # FACES: 210 # # Lines of the form # e i j L # denote an edge from vertex i to vertex j of length L. # Lines of the form # f i j k # denote a face with vertices i, j, and k. Indices are 1-based. # e e e e e e e e e e e e e e e Informs practical decisions about encoding, processing, compression, transmission,...

88 Summary Geometric Structure (2D) CONTINUOUS metric conformal structure volume form Gaussian curvature geodesic DISCRETE edge lengths length cross ratio Poincaré dual volumes angle defect more to come! Fact. There is a bijection between conformal and discrete conformal equivalence classes of metrics (via length cross ratios). Some important structures captured by discretization!

89 Thanks! DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU (J) Spring 2016